Phase-controlled pathway interferences and switchable fast-slow light in a cavity-magnon polariton system
Jie Zhao, Longhao Wu, Tiefu Li, Yu-xi Liu, Franco Nori, Yulong Liu, Jiangfeng Du
aa r X i v : . [ qu a n t - ph ] F e b Phase-controlled pathway interferences and switchable fast-slow light in acavity-magnon polariton system
Jie Zhao,
1, 2, 3, 4, ∗ Longhao Wu,
1, 2, 3, ∗ Tiefu Li,
5, 6
Yu-xi Liu, Franco Nori,
7, 8
Yulong Liu,
9, 10, † and Jiangfeng Du
1, 2, 3, ‡ Hefei National Laboratory for Physical Sciences atthe Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei 230026, China CAS Key Laboratory of Microscale Magnetic Resonance,University of Science and Technology of China, Hefei 230026, China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei 230026, China National Laboratory of Solid State Microstructures,School of Physics, Nanjing University, Nanjing 210093, China Institute of Microelectronics, Tsinghua University, Beijing 100084, China Quantum states of matter, Beijing Academy ofQuantum Information Sciences, Beijing 100193, China Theoretical Quantum Physics Laboratory, RIKEN, Saitama, 351-0198, Japan Department of Physics, The University of Michigan,Ann Arbor, Michigan 48109-1040, USA Beijing Academy of Quantum Information Sciences, Beijing 100193, China Department of Applied Physics, Aalto University,P.O. Box 15100, FI-00076 Aalto, Finland (Dated: February 25, 2021) bstract We study the phase controlled transmission properties in a compound system consisting of a 3D coppercavity and an yttrium iron garnet (YIG) sphere. By tuning the relative phase of the magnon pumping andcavity probe tones, constructive and destructive interferences occur periodically, which strongly modifyboth the cavity field transmission spectra and the group delay of light. Moreover, the tunable amplitude ratiobetween pump-probe tones allows us to further improve the signal absorption or amplification, accompaniedby either significantly enhanced optical advance or delay. Both the phase and amplitude-ratio can be usedto realize in-situ tunable and switchable fast-slow light. The tunable phase and amplitude-ratio lead to thezero reflection of the transmitted light and an abrupt fast-slow light transition. Our results confirm thatdirect magnon pumping through the coupling loops provides a versatile route to achieve controllable signaltransmission, storage, and communication, which can be further expanded to the quantum regime, realizingcoherent-state processing or quantum-limited precise measurements.
I. INTRODUCTION
Interference, due to superposed waves, plays a considerable role in explaining many classicaland quantum physical phenomena. Based on the phase-difference-induced interference patterns,ultraprecise interferometers have been created, impacting the development of modern physics andindustry [1]. In addition to the phases, waves or particles propagating through different path-ways can also introduce interference patterns. Among various types of multiple-path-inducedinterference, the Fano resonance [2] and its typical manifestations, the electromagnetically in-duced transparency (EIT) and electromagnetically induced absorption (EIABS) [3, 4], are themost well-known ones. The Fano resonance and EIT-like (or EIABS-like) line shapes are not onlyexperimentally observed in quantum systems but also in various classical harmonic-resonator sys-tems. Quantum examples include quantum dots [5], quantum wells [6], superconducting qubits [7–10], as well as Bose-Einstein condensates [11]. Classical examples [12] include coupled opticalcavities [13–16], terahertz resonators [17, 18], microwave resonators [19, 20], mechanical res-onators [21, 22], optomechanical systems [23]. However, whether in quantum or in classicalsystems, the Fano resonance, EIT- or EIABS-like spectra are normally experimentally realized ∗ These authors contributed equally to this work † [email protected] ‡ [email protected] in situ tunable and switchable absorption, transparency, and even am-plification.Meanwhile, cavity magnon polaritons in an yttrium-iron-garnet (YIG) sphere-cavity coupledsystem has attracted much attention due to its strong [35–42] and even ultrastrong couplings [43–45]. The compatibility and scalability with microwave and optical light enable magnons to be aversatile interface for different quantum devices [46–51]. At low temperatures, strong couplingbetween magnons, superconducting resonators and qubits have been demonstrated [52–56]. Sub-sequently, the EIT-like magnon-induced transparency (MIT) or the EIABS-like magnon-inducedabsorption (MIABS) of the transmitted cavity field were observed for different external couplingconditions [57]. The underlying mechanism is attributed to interferences between two transitionpathways, i.e., the direct cavity pathway and the cavity-magnon-cavity pathway, to transmit theprobe field.In addition to the coupling strength [57] and frequency detuning [58–60] between coupledmodes, phases play a vital role in wave interference control. We thus focus on the controllabil-ity of pathway interferences through the phase difference between the cavity-probe tone and themagnon-pump tone, which is introduced by the coupling loops’ technology [61–64]. The directmagnon pump is becoming useful in realizing the light-wave interface [46–48], enhancing the Kerrnonlinearity [65–67], and has also been adopted to observe the magnetostriction-induced quantumentanglement [68–72], among other applications.Together with the cavity-probe tone, a magnon-pump tone introduces a controllable relativephase to the system, and thus the path interference can be real-time controlled. Changing the two-tone phase difference, we can switch the cavity-probe spectra from the original magnon-inducedtransparency instantly to the magnon-induced absorption, or even the Fano line shape . Further-more, the tunable pump-probe amplitude ratio allows us to further improve the signal absorption,transparency, or amplification, accompanied by a significant enhancement by nearly agnitude of the optical advance or delay time compared to the case without magnon pump [57].In particular, the tunable phase and amplitude ratio also lead to the zero reflection of the trans-mitted light, which is accompanied by an abrupt transition of delay time. Our results confirmthat direct magnon pumping provides a versatile route to control signal transmission, storage, andcommunication, and can be further expanded to coherent state processing in the quantum regime. -150 -75 0 75 150 p (MHz) -4-20 S ( d B ) -150 -75 0 75 150 p (MHz) -8-6-4-2 S ( d B ) Port 1 Port 2
QIL R , A B
Splitter IQ MixerCirculatorVNA AWG
CavityMagnon g Cavity g Magnon
HighLow (a)(b) (c)(d) xyz D e s t r u c t i v e C tt i IG. 1. Measurement setup and phase-induced interference mechanism diagrams. (a) The system consistingof a three-dimensional (3D) copper cavity and a YIG sphere, which is coherently pumped by the couplingloops shown as a black coil surrounding the YIG sphere. The red arrows and colors indicate the magneticfield directions and amplitudes of the TE mode distribution, respectively. The YIG sphere is placed atthe area with maximum magnetic field distribution inside a 3D copper cavity box to obtain a strong cavity-magnon coupling. A small hole at the cavity sidewall is assembled with a standard SubMiniature version Aconnector (SMA connector), allowing us to do the reflection measurement S of the probe field, i.e., such aSMA connector works as both the signal input and readout port. A beam of coherent microwave comes outfrom port 1 of the vector network analyzer (VNA) and splits into two beams, working as the magnon-pumptone and the cavity-probe tone. Here, we use an in-phase and quadrature mixer (I-Q mixer) and an arbitrarywaveform generator (AWG) to control and tune the phase difference ϕ and pump-probe amplitude ratio δ = ε m /ε c between the pump and probe tones. The interfering results are extracted by the circulator andfinally transferred to port 2 of the VNA. (b) Diagram showing the relative phase between the magnon pumpand cavity probe in the cavity-magnon coupled system. (c) The corresponding energy-level diagram. Twotransition pathways to the higher energy level: 1 (cid:13) probe-tone-induced direct excitation, and 2 (cid:13) pump-toneexcites magnons and then coherently transfers there to cavity photons. (d) Measurements of the reflectionspectra S versus the detuning ∆ p = ω c − ω p = ω m − ω p . The relative phase difference between pumpand probe tones can be developed to realize an in situ switchable constructive and destructive interference,presented as MIABS with ϕ = 0 . π, δ = 1 . and MIT with ϕ = 1 . π, δ = 1 . . II. EXPERIMENTAL SETUP
As shown in Fig. 1(a), our system consists of a 3D copper (Cu) cavity with an inner dimensionof × × mm and an YIG sphere with a 0.3 mm diameter. A static magnetic field H static applied in the x - y plane tunes the magnon frequency. The simulated cavity-mode magnetic fielddistribution is shown at the bottom of Fig. 1(a), where the arrows and colors indicate the cavitymode magnetic field directions and amplitudes. The YIG sphere is placed near the magnetic fieldantinode of the cavity TE mode. The magnetic components (along the z axis) of the microwavefield at this antinode is perpendicular to the static magnetic bias field.Here, we are only interested in the low excited states of the Kittel mode, in which all the spinsprecess in phase. Under the Holstein-Primakoff transformation, such collective spin mode can be5implified to a harmonic resonator, which introduces the magnon mode. In our setup, the cavitymode couples to the magnon mode with coupling strength g = 7 . , which is larger than themagnon decay rate κ m = 1 . , but smaller than the cavity decay rate κ c = 113 . .In our experiment, a beam of coherent microwave is emitted from port 1 of a VNA and thendivided through a splitter into two beams, one of which is used to probe the cavity (probe tone)and another beam is used to pump the magnon (pump tone) by incorporating the coupling looptechnique, which is schematically shown in the dashed rectangle of Fig. 1(a). The probe tone isinjected into the cavity through antenna 1, which induces the cavity external decay rate κ c1 =21 . . The pump tone is injected through antenna 2, which introduces the magnon externaldecay rate κ m1 = 0 . . Note that the phase ϕ c = 0 and amplitude ε c of the probe tone arefixed (i.e., working as a reference), and the phase ϕ and amplitude ε m of the magnon-pump toneare tunable and controlled by an arbitrary wave generator with an in-phase and quadrature mixer(I-Q mixer). III. MODEL
By considering the cavity-magnon coupling, as well as the pump and probe tones [model inFig. 1(b)], the system Hamiltonian becomes H = ω c a † a + ω m m † m + g ( a † m + m † a )+ i p η c κ c ε c (cid:0) a † e − iω p t − ae iω p t (cid:1) + i p η m κ m ε m (cid:0) m † e − iω p t − iϕ − me iω p t + iϕ (cid:1) . (1)Here, a † ( a ) and m † ( m ) are the creation (annihilation) operators for the microwave photon andthe magnon at frequencies ω c and ω m , respectively, and we choose units with ~ = 1 . The magnonfrequency ω m linearly depends on the static bias field H static and is tunable within the range of afew hundred MHz to about 45 GHz; ε c ( ε m ) is the microwave amplitude applied to drive the cavity(magnon). Here, we introduce the coupling parameter η c = κ c1 /κ c , (2) η m = κ m1 /κ m (3)6o classify the working regime of the cavity (the magnon). The parameter η c ( η m ) classifies threeworking regimes for the cavity (magnon) into three types: overcoupling regime for η c ( η m ) > / ;critical-coupling regime for η c ( η m ) = 1 / ; and undercoupling regime for η c ( η m ) < / . In ourexperiment, the cavity works in the undercoupling regime ( η c < / ) and the magnon works inthe critical coupling regime ( η m = 1 / ).Experimentally, the reflection signal from the cavity is circulated and then transferred to port2 of the VNA to carry out the spectroscopic measurement, which corresponds to the steady-statesolution of the Hamiltonian Eq. (1). The transmission coefficient t p of the probe field is defined asthe ratio of the output-field amplitude ε out to the input-field amplitude ε c at the probe frequency ω p : t p = ε out /ε c . With the input-output boundary condition, ε out = ε c − p η c κ c h a i , (4)we can solve the transmission coefficient t p of the probe field as [73] t p = t probe + t pump , (5)with t probe = 1 − η c κ c ( i ∆ p + κ m )( i ∆ p + κ c ) ( i ∆ p + κ m ) + g , (6) t pump = ig √ η c κ c √ η m κ m δe − iϕ ( i ∆ p + κ c ) ( i ∆ p + κ m ) + g . (7)Here ∆ p is the detuning between the probe frequency ω p and either the cavity resonant frequency ω c or the magnon frequency ω m . In our experiment, the cavity is resonant with the cavity, i.e., ∆ p = ω c − ω p = ω m − ω p ; (8)and δ = ε m /ε c (9)is the pump-probe amplitude ratio. Equation (5) clearly shows that the transmission coefficientcan be divided into two parts:1. t probe in Eq. (6), the contribution from the cavity-probe tone, represents the traditionalpathway-induced interference;2. t pump in Eq. (7), the contribution from the magnon-pump field, affects the interference andmodifies the transmission of the probe field.7s shown in Fig. 1(c), there exist two transition pathways for the cavity: the probe-tone-induceddirect excitation, and the photons transferred from magnon excitations. When the cavity decayrate (analog to broadband of states) is much larger than the magnon decay rate (analog to a nar-row discrete quantum state in other quantum systems), Fano interference happens and has beensuccessfully used to explain the MIT and MIABS phenomenon in cavity magnon-polariton sys-tems [57]. Besides pathway-induced interference, the steered phase ϕ of the wave provides anotheruseful way to generate and especially control the interferences, as shown in Fig. 1(d).We emphasize that in this paper we focus on how the phase difference ϕ and pump-probe ratio δ = ε m /ε c affect the interference, and we explore its potential applications, such as controllablefield transmission and in situ switchable slow-fast light . The S spectrum and group-time delaymeasurement are carried out on the VNA and then fitted by T = | t p | (10)and τ = − ∂ [arg ( t p )] ∂ ∆ p , (11)respectively. IV. PHASE INDUCED INTERFERENCE AND CONTROLLABLE MICROWAVE FIELD TRANS-PORT
We first study how the phase of the magnon-pump tone affects the transmission of the cavity-probe field. In Fig. 2 (a), we present experimental results of the transmission, when the pump-probe ratio is δ = ε m /ε c = 1 . . In this setup, the phase ϕ is continuously increased from 0 to 2 π using an I/Q mixer, and is shown in the x axis of Fig. 2 (a). Then we conduct the S measurementsand the recorded spectra are plotted versus the detuning frequencies ∆ p . The colors represent therelative steady-state output amplitude (in dB units) at different frequency and pump-probe ratios.Figure 2(a) shows that the interference mainly happens around ∆ p = 0 and can be controlled insitu by changing the phase ϕ .As shown in Fig. 2(b), where ϕ is set to . π , destructive interference happens and an obviousdip appears around ∆ p = 0 . This behavior can be regarded as MIABS. However, if we set ϕ = heoryExperiment -150 75 0 75 150 (a)(b) -6.5-13 0-4.5-92-1.5-5 1-2.5-6-150 150 -150 -75 15075 FIG. 2. S spectrum versus relative phase difference ϕ . (a) Measured transmission spectrum S versusphase ϕ and detuning ∆ p . The colors indicate the transmitted amplitudes in dB units. (b) Measured outputspectrum S with phases: 1 (cid:13) ϕ = 0 . π , 2 (cid:13) ϕ = 0 . π , 3 (cid:13) ϕ = 1 . π , and 4 (cid:13) ϕ = 1 . π . Here, thepump-probe amplitude ratio is fixed at δ = 1 . . Red-solid lines are the corresponding theoretical results. . π , constructive interference happens and an obvious amplification window appears around ∆ p = 0 . This behavior can be described as magnon-induced amplification (MIAMP). When ϕ is set to . π or . π , sharp and Fano-interference-like asymmetry spectra are observed evenwhen the cavity and magnon are exactly resonant.Although the interference originates from the coherent cavity-magnon coupling, Fig. 2 clearlyshows that the phase ϕ plays a key role in realizing an in situ tunable and controllable interfer-ence (e.g., constructive or destructive interference) , which can be further engineered to control theprobe-field transmission. Note that in previous studies [57] MIABS was only observed in the cav-9ty overcoupling regime (i.e., η a > / ) and MIT was only observed in the cavity undercouplingregime (i.e., η a < / ). In contrast to this, here we realize a phase-dependent and switchable MI-ABS and MIT, as well as MIAMP in a fixed undercoupling regime ( η c = 0 . in our experiment).We emphasize that the destructive interference-induced MIABS is a unique result of phase mod-ulation. The observed asymmetric Fano line shapes could be useful to realize Fano-interferencesensors or precise measurements, using the magnon-pump method realized in our work. V. AMPLITUDE RATIO OPTIMIZED MAGNON-INDUCED-ABSORPTION
Recall the magnon-pump transmission coefficient t pump in Eq. (7). There, the phase ϕ deter-mines the type of interference, e.g., constructive or destructive. However, the pump-probe ratio δ = ε m /ε c also affects the degree of interference, and thus can be used to control the probe-fieldtransmissions t p . As shown in Fig. 3(a), a color map is used to present the experiment results.Along the x axis, the amplitude ratio δ is continuously increased from 0 to 6.5, by changing theoverall voltage amplitude applied to the I and Q ports of an I-Q mixer. Then we conduct the S measurements and the steady-state output-field amplitudes are plotted versus the frequency de-tuning ∆ p . The colors in Fig. 3(a) represent the relative strength of the steady-state output field(in dB units) at a different frequency. Here, the chosen phase ϕ = 0 . π results in MITs when δ < . , while MIABSs dominate the output response in the regime δ > . . We then studyhow the pump-probe ratio δ affects the central absorption window of the S spectra.Figure 3(a) shows that interference occurs around ∆ p = 0 and is in situ controlled by changingthe pump-probe ratio δ . The center blue-colored area represents an ideal absorption (transmission T < . ) of the probe field.Figure 3 (b) shows the extreme values of the transmission coefficients around ∆ p = 0 versusthe pump-probe ratio δ . In the yellow area, we find the local maximum values of the MITs, andthe local minimum values are found for MIABSs in the blue area. An obvious dip appears around δ = 3 and the minimum transmission value is less than 1% (voltage amplitude ratio), whichcorresponds to an optimized and ideal probe-field absorption.Figure 3(c) shows the evolution process from MIT to MIABS by gradually increasing thepump-probe ratio δ . When δ = 0 , corresponding to case 1 (cid:13) of Fig. 3(c), our scheme recoversthe traditional MIT case when no magnon pump is applied. When the magnon pump is introducedand its strength is continuously increased, the transparency window disappears and is replaced by10
150 -75 0 75 150
Theory -150 -75 0 75 150
Experiment (a) ((cid:0)(cid:1) E x t r e m e A m p li t u d e ( d B ) -1.0-2.5-4.00-20-40-600 2 4 6-1.0-2.5-4.00-20-42-150 -75 150 1.0-4.0-9.0-150 -75 150 FIG. 3. Measured transmission spectrum S versus pump-probe amplitude ratio δ with phase fixed at ϕ = 0 . π . (a) Measured output spectrum versus amplitude ratio δ and detuning ∆ p . The colors indicatetransmitted power in dBs. (b) The extreme values of the S transmission spectra of the output field versusthe amplitude ratio parameter δ . In the light-yellow (light-blue) regime, the extreme values represent themaximum (minimum) transmission amplitudes of the peaks (dips) around ∆ p = 0 . (c) Measured transmis-sion spectrum S with amplitude ratio: 1 (cid:13) δ = 0 , 2 (cid:13) δ = 0 . , 3 (cid:13) δ = 3 . , and 4 (cid:13) δ = 5 . . Red-solid linesare the corresponding theoretical results. an obvious absorption dip, as shown in cases 2 (cid:13) and 3 (cid:13) of Fig. 3(c). With an even larger pump-probe ratio, the MIABS dips become asymmetry gradually, such as the spectrum in the case 4 (cid:13) ofFig. 3(c). Comparing with other results in Fig. 3(c), we can find that the experimental data do nofit so well with the theory in case 4 (cid:13) of Fig. 3(c). This is induced by the additional cavity-antenna11 coupling. Due to the existence of this tiny coupling, the magnon pump signal also pumps thecavity. With a modest magnon-pump strength, the additional cavity pump does not affect the sys-tem seriously, so that the theory fit the experiment data well. With a relatively strong magnonpump, the side effects of the additional cavity pump become larger, though it does not change theline shape. Therefore, the experiment data and theory do not fit so well when the magnon pump isrelatively strong [73]. Similar phenomena can also be observed in the case 4 (cid:13) of Fig. 4(c).We emphasize one main result of this paper: the absorption dips appear with an under-couplingcoefficient of η a = 0 . in our experiment. However, absorptions only happen in the overcouplingregime in traditional cases . Moreover, Figs. 3(a) and (c) show that δ can be used to switch thetransmission behavior from the magnon-induced transparency to the magnon-induced absorption .Note that the type of interference, destructive interference or constructive interference, dependson the value of the phase ϕ . However, the interference intensity is determined and optimized bythe pump-probe ratio δ . As shown in Fig. 3(c), the dip of S is 42 dB lower than the baseline.The dip amplitude is quite close to zero, which indicates that a zero reflection is generated by thedestructive interference. VI. AMPLITUDE RATIO OPTIMIZED MAGNON-INDUCED-AMPLIFICATION
We now study how the amplitude ratio of δ = ε m /ε c affects the MIAMP. In this case, the phaseis fixed at ϕ = 1 . π , where constructive interference dominates the transmission of the outputfield. As shown in Fig. 4(a), a color map is used to present the measurement results. Along the x axis, the pump-probe ratio δ is continuously increased from 0 to 6.5. Then we conduct the S measurement, and the steady-state transmission spectra are plotted versus the frequency detuningparameter ∆ p . The colors in Fig. 4(a) represent the transmission amplitudes of the steady-stateoutput field (in dB units) at different frequencies. We then study how the amplitude δ affects thecenter amplification window of the S spectra.Figure 4(a) clearly shows that constructive interference happens around ∆ p = 0 and are in situ controlled by changing the pump-probe ratio δ . Magnon-pump-induced constructive interferencehappens when the probe field is nearly resonant with the cavity (also the magnon), and amplifi-cation windows appear. Around ∆ p = 0 , the color changes from light blue to orange when thepump-probe ratio δ increases from 0 to 6.5. This indicates that the higher amplification can beobtained with a larger pump-probe ratio δ . 12 E x t r e m e A m p li t ude ( d B ) Theory
Experiment (c)
Experiment
Theory (b) (a) -150 -75 0 75 150-4.0-2.00-150 -75 0 75 150-4.004.0-150 -50 0 75 150-4-2.5-1-150 -75 0 75 150-4.0-2.00
FIG. 4. Measured transmission spectrum S versus pump-probe amplitude ratio δ = ε m /ε c with phasefixed at ϕ = 1 . π . (a) Measured output spectra S versus amplitude ratio δ and frequency detuning ∆ p . The colors indicate the transmitted amplitude in dB units. (b) The extreme values of the S trans-mission spectra of the output field versus the amplitude-ratio parameter δ . The extreme values representthe maximum transmission amplitude of the peaks around ∆ p = 0 . (c) Measured transmission spectra S with amplitude ratios: 1 (cid:13) δ = 0 , 2 (cid:13) δ = 0 . , 3 (cid:13) δ = 1 . , and 4 (cid:13) δ = 4 . . The red-solid lines are thecorresponding theoretical results. Figure 4(b) shows how the peak values in the amplification window change versus the ampli-tude ratio δ . The amplification coefficient is monotonously dependent on the increment of thepump-probe ratio δ . Although the maximum pump-probe ratio is δ = 6 . in our experiment, weemphasize that a higher transmission gain can be obtained using a larger pump power.13igure 4(c) clearly shows the evolution of the transmission spectrum from MIT to MIAMPwhen we gradually increase the pump-probe ratio δ . When δ < . , an obvious transparencywindow appears. When δ = 1 . , the peak value of the transparency window equals the value ofthe baseline, showing the ideal MIT phenomenon. Further increasing the pump strength, we canobserve MIAMP. When δ = 4 . , an obvious amplification window appears, producing MIAMP.Note that the phase is fixed at ϕ = 1 . π to produce constructive interference. When theamplitude ratio is set to δ = 0 , i.e., no magnon pump, our scheme also recovers the traditionalcase without a magnon pump and only MIT is observed. This result is, of course, the same ascase 1 (cid:13) in Fig. 3(c). We point out another main result that the pump-probe ratio δ can be usedto realize and control the magnon-induced amplifications . Figures 4(a) and 4(c) show that δ canbe used to switch the system response from MIT to MIAMP. Note that the interference type, suchas constructive interference discussed here, depends on the value of the phase ϕ ; however, theinterference intensity is determined and optimized by the pump-probe ratio δ . VII. SWITCHABLE FAST- AND SLOW-LIGHT BASED ON THE PHASE AND AMPLITUDERATIO
The group delay or advance of light always accompanies EIT or EIABS. In this experiment,we show that the group delay (slow light) and group advance (fast light) can also be realized inour cavity magnon-polariton system. Similar to the discussions above, the phase ϕ is the keyparameter that determines the interference type, e.g., destructive or constructive. Therefore, thephase ϕ provides a tunable and in situ switched group advance or delay of the probe field. Theextreme values of the delay time are measured and presented in Fig. 5, choosing the same phases ϕ = 0 . π and ϕ = 1 . π , which are also used in Figs. 3 and 4, respectively.In Fig. 5(a), the phase is set to ϕ = 0 . π . When we increase the pump-probe ratio δ , a longeradvance time is achieved, but immediately changes to time delay when δ > . . Further increasing δ reduces the delay time. In Fig. 5(c), we present the phase of transmission signals at differentprobe frequencies with δ = 2 . (case 1 (cid:13) ) and δ = 3 . (case 2 (cid:13) ). The phase changes drasticallyaround ∆ p = 0 with opposite directions. The drastic changes of the phase result in a long advanceor delay time, while the phase-change direction reversal results in the sharp transition from timeadvance to time delay. Accompanying the sharp transition in Fig. 5(a), we observe the longesteither delay or advance times. Therefore, the pump-probe ratio δ allows to optimize and switch the E x t r e m e D e l a y T i m e ( n s ) P ha s e (c) -50 500 -50 500-101 P h a s e ( r a d ) -440 FIG. 5. Measured time delay versus pump-probe ratio δ for the phase ϕ = 0 . π (a); and ϕ = 1 . π (b).Light-yellow area indicates the group-delay regime, and the light-blue area indicates the group-advanceregime. (c) Measured unwrapped phase versus frequency detuning ∆ p with δ = 2 . [point 1 (cid:13) in (a)] and δ = 3 . [point 2 (cid:13) in (a)] for ϕ = 0 . π . probe microwave from fast to slow light, or inversely . Comparing the abrupt transition in Fig. 5(a)with the zero reflection discussed in Sec. V, we find that the delay time abrupt transition and thezero reflection occur at the same parameter setup. It is notable that the discontinuity and abrupttransition are always accompanied by the zero reflection in coupled resonator systems. In Fig. 5(b),we set the phase to ϕ = 1 . π and mainly observe constructive interference. In this case, the timedelay monotonously increases with the pump-probe ratio δ . Note that the pump-probe ratio usedin Fig. 5(b) is not its limitation, therefore longer delay times can be achieved by further increasing δ . Figure 5 also shows that when the amplitude ratio δ ≤ . , the delay time is a negative numberwhich corresponds to fast light with ϕ = 0 . π , and the positive delay time corresponds to slow15 ABLE I. Summary of MIT, MIABS, MIAMP and Fano resonance observed experimentally for differentvalues in parameter space. Amplitude Ratio δ > . Phase ϕ . π MIT NULL MIABS MIABS MIABS Fano . π MIT MIT MIT MIT (perfect) MIAMP MIAMP light with ϕ = 1 . π . Thus the phase parameter ϕ can also be used to switch fast and slowlight. When δ = 0 , i.e., no magnon pump, our scheme recovers the traditional MIT and only a16-ns delay time is achieved. By applying the magnon pump and optimizing ϕ and δ , the timedelay, as well as advance, can be enhanced by nearly 2 orders of magnitude compared with thecase without magnon pump . For our scheme, the pump-probe amplitude ratio and phase differencemediated path interference can result in the zero reflection, which is accompanied with a delay timeabrupt transition. In our experiment, Fig. 5(a) clearly shows such an abrupt transition and greatlyenhanced fast-slow light around this point. We can find that the experimental data deviates fromthe theoretical result around the abrupt transition. This is mainly induced by the imperfect systemsetups, such as limited output precision of AWG, imperfectness of the I-Q mixer and unstablemagnon frequency [73]. VIII. CONCLUSION
We experimentally study how the magnon pump affects the probe-field transmission, and theobserved results are summarized in Table. I. Two parameters, the relative phase ϕ and the pump-probe ratio δ between pump and probe tones, are studied in detail. The main results of this workare as follows:• the unconventional MIABS of the transmitted microwave field is observed with the cavityin the undercoupling condition;• MIAMP phenomena is realized in our experiment;• asymmetric Fano-resonance-like spectra are observed even when the cavity is resonant withthe magnon; 16 by tuning the phase of the magnon pump, we can easily switch between MIT, MIABS andMIAMP;• by tuning the pump and probe ratio, the MIABS and MIAMP can be further optimized,accompanied by greatly enhanced advanced or slow light by nearly 2 orders of magnitude;• the tunable phase and amplitude ratio can lead to the zero reflection of the transmitted lightand abrupt fast-slow light transitions.;• both the ϕ and δ can be used to carry out the in situ switch of fast and slow light.Our results confirm that direct magnon pumping through the coupling loops provides a versa-tile route to achieve controllable signal transmission, storage, and communication, which can befurther expanded to coherent state processing in the quantum regime. Furthermore, by exploitingmulti-YIG spheres or multimagnon modes systems, the amplification or absorption bandwidth canbe increased, resulting in a broadband coherent signal store device. The sharp peak and asymmet-ric Fano line shape indicate that our platform has great potential in the application of high-precisionmeasurement of weak microwave fields [74, 75]. Our two-tone pump scheme and phase-tunableinterference can also be accomplished in other coupled-resonator systems, such as optomechanicalresonators, which explores effects of mechanical pump on light transmission [76–84], and evenin circuit-QED systems, in which photon transmission can be controlled through a circuit-QEDsystem [85–88]. ACKNOWLEDGMENTS
This work is supported by the National Key R&D Program of China (Grant No. 2018YFA0306600),the CAS (Grants No. GJJSTD20170001 and No. QYZDY-SSW-SLH004), Anhui Initiative inQuantum Information Technologies (Grant No. AHY050000), and the Natural Science Foun-dation of China (NSFC) (Grant No. 12004044). F.N. is supported in part by: NTT Research,Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technol-ogy Agency (JST) (via the CREST Grant No. JPMJCR1676), Japan Society for the Promotionof Science (JSPS) (via the KAKENHI Grant No. JP20H00134 and the JSPS-RFBR Grant No.JPJSBP120194828), the Asian Office of Aerospace Research and Development (AOARD), andthe Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06.17ote added – Recently, we become aware of a study presenting an infinite group delay andabrupt transition in a magnonic non-Hermitian system [33]. [1] M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor,A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock,
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